A bounded linear operator A on a complex Hilbert space H is a <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44146_2025_208_Article_IEq1.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\rho $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ρ</mi> </math></EquationSource> </InlineEquation>-contraction (<InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44146_2025_208_Article_IEq7.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$\rho >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ρ</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>) if there is a unitary operator U on space K containing H such that <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44146_2025_208_Article_IEq8.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="117" /> </InlineMediaObject> <EquationSource Format="TEX">$A^n=\rho P_HU^n|H$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>A</mi> <mi>n</mi> </msup> <mo>=</mo> <mi>ρ</mi> <msub> <mi>P</mi> <mi>H</mi> </msub> <msup> <mi>U</mi> <mi>n</mi> </msup> <mrow> <mo stretchy="false">|</mo> <mi>H</mi> </mrow> </mrow> </math></EquationSource> </InlineEquation> for all <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44146_2025_208_Article_IEq9.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$n\ge 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44146_2025_208_Article_IEq10.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">$P_H$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>P</mi> <mi>H</mi> </msub> </math></EquationSource> </InlineEquation> denotes the orthogonal projection from K onto H. U is called a unitary <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44146_2025_208_Article_IEq1.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\rho $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ρ</mi> </math></EquationSource> </InlineEquation>-dilation of A. Such operators originate from the early work of Sz.-Nagy and Foiaş (1966) and generalize the usual contractions and numerical contractions. In this survey, we briefly describe their properties, covering (1) their characterizations, (2) the related <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44146_2025_208_Article_IEq1.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\rho $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ρ</mi> </math></EquationSource> </InlineEquation>-radius, defined as <InlineEquation ID="IEq13"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44146_2025_208_Article_IEq13.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="203" /> </InlineMediaObject> <EquationSource Format="TEX">$w_{\rho }(A)=\inf \{\lambda >0 : (1/\lambda )A$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>w</mi> <mi>ρ</mi> </msub> <mrow> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mo movablelimits="true">inf</mo> <mo stretchy="false">{</mo> <mi>λ</mi> <mo>></mo> <mn>0</mn> <mo>:</mo> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mrow> <mi>A</mi> </mrow> </mrow> </math></EquationSource> </InlineEquation> is a <InlineEquation ID="IEq14"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44146_2025_208_Article_IEq1.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\rho $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ρ</mi> </math></EquationSource> </InlineEquation>-contraction<InlineEquation ID="IEq15"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44146_2025_208_Article_IEq15.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\}$</EquationSource> <EquationSource Format="MATHML"><math> <mo stretchy="false">}</mo> </math></EquationSource> </InlineEquation>, (3) their similarity to contractions, (4) properties of their powers on a fixed vector, and (5) their unitary <InlineEquation ID="IEq16"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44146_2025_208_Article_IEq1.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\rho $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ρ</mi> </math></EquationSource> </InlineEquation>-dilations.

A brief survey on $\rho $-contractions and unitary $\rho $-dilations

期刊论文
发表日期：

2025年11月18日

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Pei Yuan Wu,
Hwa-Long Gau

摘要

A bounded linear operator A on a complex Hilbert space H is a $\rho $ $ρ$ -contraction ( $\rho >0$ $ρ > 0$ ) if there is a unitary operator U on space K containing H such that $A^n=\rho P_HU^n|H$ $A^{n} = ρ P_{H} U^{n} | H$ for all $n\ge 1$ $n \geq 1$ , where $P_H$ $P_{H}$ denotes the orthogonal projection from K onto H. U is called a unitary $\rho $ $ρ$ -dilation of A. Such operators originate from the early work of Sz.-Nagy and Foiaş (1966) and generalize the usual contractions and numerical contractions. In this survey, we briefly describe their properties, covering (1) their characterizations, (2) the related $\rho $ $ρ$ -radius, defined as $w_{\rho }(A)=\inf \{\lambda >0 : (1/\lambda )A$ $w_{ρ} (A) = inf {λ > 0 : (1 / λ) A$ is a $\rho $ $ρ$ -contraction $\}$ $}$ , (3) their similarity to contractions, (4) properties of their powers on a fixed vector, and (5) their unitary $\rho $ $ρ$ -dilations.

A brief survey on \(\rho \)-contractions and unitary \(\rho \)-dilations

摘要